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Master Production Scheduling
Material Requirements Planning
Order Scheduling
Process Planning
Strategic Capacity Planning
Aggregate Planning
Long-range
Intermediate-range
Short-range
Hierarchy of Planning Problems
Ch 11 - 22© 1998 by Prentice-Hall IncRussell/Taylor Oper Mgt 2/e
Hierarchical Planning ProcessItems
Product lines or families
Individual products
Components
Manufacturing operations
Resource level
Plants
Individual machines
Critical work centers
Production Planning Capacity Planning
Resource Requirements Plan
Rough-Cut Capacity Plan
Capacity Requirements Plan
Input/Output Control
Aggregate Production Plan
Master Production Schedule
Material Requirements Plan
Shop Floor Schedule
All work centers
Forecast of AggregateDemand For t Period
Planning Horizon
Forecast of AggregateDemand For t Period
Planning Horizon
Aggregate Production Plan Determination of Aggregate Production &
Work Force Levels for t Period Planning Horizon
Aggregate Production Plan Determination of Aggregate Production &
Work Force Levels for t Period Planning Horizon
Master ProductionSchedule
Aggregate Planning Translating annual or quarterly business plans
into broad labor and output plans for the intermediate term (6 to 18 months).
Objective: to minimize the cost of resources required to meet demand over that period.
Optimal combination of production rate, workforce level, and inventory on hand is sought throughout the intermediate term.
Ch 11 - 7© 1998 by Prentice-Hall IncRussell/Taylor Oper Mgt 2/e
Inputs and Outputs to Aggregate Production Planning
AggregateProductionPlanning
CompanyPolicies
FinancialConstraints
StrategicObjectives
Units or dollarssubcontracted,backordered, or
lost
CapacityConstraints
Size ofWorkforce
Productionper month
(in units or $)
InventoryLevels
DemandForecasts
Ch 11 - 6© 1998 by Prentice-Hall IncRussell/Taylor Oper Mgt 2/e
Aggregate Production Planning (APP)
Matches market demand to company resources
Plans production 6 months to 12 months in advance
Expresses demand, resources, and capacity in general terms
Develops a strategy for economically meeting demand
Establishes a companywide game plan for allocating resources
Goal: To plan gross work force and production levels and set firm-wide production plans.
We try to determine aggregate production targets and the levels of resources required to achieve these production goals given the demand projection for the intermediate term.
Namely given demand as D1, D2,…,DT
Find the number of workers employed in each period
Find the number of aggregate units to be produced in each period
Concept is based on the idea of an “aggregate unit” of production. Aggregate Planning requires aggregation of different goods or services.
They may be Actual units of production if items are similar Weight (tons of steel) Volume (gallons of gasoline) Dollar value (Value of sales) Fictitious aggregate units
Aggregation Unit - Example Six different models of washing machines
Model Number Number of Worker-Hours Required to Produce
Selling Price
Percentage of total sales
A5532 4.2 $285 32
K4242 4.9 345 21
L9898 5.1 395 17
L3800 5.2 425 14
M2624 5.4 525 10
M3880 5.8 725 6
Can we use dollars as an aggregation unit? Notice: Price is not necessarily proportional to worker hours (i.e., cost): why?Use a fictitious washing machine as an aggregation unit which requires(.32)(4.2)+(.21)(4.9)+(.17)(5.1)+(.14)(5.2)+(.10)(5.4)+(.06)(5.8)=4.856 hours of labor timeForecasts for demand for aggregate units can be obtained by taking a weighted average (using the same weights) of individual item forecasts.
Different levels of aggregation
Items: Final products to be delivered to the customer Families: A group of items that a share common
manufacturing setup cost Types: Groups of families with production quantities
that are determined by a single aggregate production plan
Above may not always work, the aggregation method should be consistent with the firm’s organizational structure, product line, planning needs and availability of forecast and other data.
Production Rate
What is the best production curve in seasonal business
Demand
Time
Ch 11 - 8© 1998 by Prentice-Hall IncRussell/Taylor Oper Mgt 2/e
Strategies for Meeting Demand1. Use inventory to absorb fluctuations in demand
(level production)
2. Hire and fire workers to match demand (chase demand)
3. Maintain resources for high demand levels
4. Increase or decrease working hours (over & undertime)
5. Subcontract work to other firms
6. Use part-time workers
7. Provide the service or product at a later time period (backordering)
Ch 11 - 9© 1998 by Prentice-Hall IncRussell/Taylor Oper Mgt 2/e
Strategy Details
Level production - produce at constant rate & use inventory as needed to meet demand
Chase demand - change workforce levels so that production matches demand
Maintaining resources for high demand levels -ensures high levels of customer service
Overtime & undertime - common when demand fluctuations are not extreme
Ch 11 - 10© 1998 by Prentice-Hall IncRussell/Taylor Oper Mgt 2/e
Strategy Details
Subcontracting - useful if supplier meets quality & time requirements
Part-time workers - feasible for unskilled jobs or if labor pool exists
Backordering - only works if customer is willing to wait for product/services
Ch 11 - 11© 1998 by Prentice-Hall IncRussell/Taylor Oper Mgt 2/e
Level Production
Time
Production
Demand
Units
Ch 11 - 12© 1998 by Prentice-Hall IncRussell/Taylor Oper Mgt 2/e
Chase Demand
Time
Units
Production
Demand
Ch 11 - 23© 1998 by Prentice-Hall IncRussell/Taylor Oper Mgt 2/e
Aggregate Planning for Services
1. Most services can’t be inventoried
2. Demand for services is difficult to predict
3. Capacity is also difficult to predict
4. Service capacity must be provided at the appropriate place and time
5. Labor is usually the most constraining resource for services
Overview of the Problem
Suppose that D1, D2, . . . , DT are the forecasts of demand for aggregate units over the planning horizon (T periods.)
The problem is to determine both work force levels (Wt) and production levels (Pt ) to minimize total costs over the T period planning horizon.
Important Issues Smoothing. Refers to the costs and disruptions
that result from making changes from one period to the next.
Bottleneck Planning. Problem of meeting peak demand because of capacity restrictions.
Planning Horizon. Assumed given (T), but what is “right” value? Rolling horizons and end of horizon effect are both important issues.
Treatment of Demand. Assume demand is known. Ignores uncertainty to focus on the predictable/systematic variations in demand, such as seasonality.
Costs in Aggregate Planning
Smoothing costs changing size of the work force changing number of units produced
Holding costs: primary component: opportunity cost of investment
Shortage costs: Cost of demand exceeding stock on hand.
Regular time costs Overtime or subcontracting costs Idle time costs
Smoothing costs
Holding and Back-Order Costs
Back-orders Positive inventory
Slope = CP
Slope = Ci
$ C
ost
Inventory
Prototype Problem Forecast demands over the next six
months for disk drives.Month Forecast
January 1280
February 640
March 900
April 1200
May 2000
June 1400
*Inventory at the end of December expected to be 500*Company requires ending inventory of 600 at the end of June*Initial workforce is 300
Cost of hiring one worker = CH = $500Cost of firing one worker = CF = $1000Inventory holding cost per unit, per month = CI = $80
Prototype Problem Plant manager observed that over 22
working days, with the workforce level of 76, the firm produced 245 disk drives.
K= # of aggregate units produced per worker per day = 245 / (22 * 76) = 0.14653
Month Net Predicted Demand
Net Predicted Cumulative Demand
January 780 780
February 640 1420
March 900 2320
April 1200 3520
May 2000 5520
June 2000 7520
Zero Inventory Plan (Chase)
Month Number of working days
Number of units produced per worker
Predicted net demand
Minimum number of workers required
January 20 2.931 780 267
February
24 3.517 640 182
March 18 2.638 900 342
April 26 3.810 1200 315
May 22 3.224 2000 621
June 15 2.198 2000 910
Zero Inventory Plan (Chase)Month Number
of workers
Number hired
Number fired
Number of units per worker
Number of units produced
Cumulative production
Cumulative demand
Ending Inventory
January 267 33 2.931 783 783 780 3February
182 85 3.517 640 1423 1420 3
March 342 160 2.638 902 2325 2320 5April 315 27 3.810 1200 3525 3520 5May 621 306 3.224 2002 5527 5520 7June 910 289 2.198 2000 7527 7520 7Total 755 145 30
Hiring costs = 755 * 500 = $377,500Firing costs = 145 * 1000 = $145,000Holding costs = (600+30) * 80 = $50,400Total cost = $ 572,900
Constant workforce plan (Level)
Month Cumulative net demand
Cumulative number of units produced per worker
Ratio (rounded up)
January 780 2.931 267
February 1420 6.448 221
March 2320 9.086 256
April 3520 12.896 273
May 5520 16.120 343
June 7520 18.318 411
Constant workforce plan (Level)Month Number of
units per worker
Number of units produced
Cumulative production
Cumulative demand
Ending Inventory
January 2.931 1205 1205 780 425February 3.517 1445 2650 1420 1230March 2.638 1084 3734 2320 1414April 3.810 1566 5300 3520 1780May 3.224 1325 6625 5520 1105June 2.198 903 7528 7520 8Total 5962
Hiring costs = (411-300) * 500 = $55,500 Holding costs = (5962+600) * 80 = $524,960 Total cost = $ 580,460
Mixed Strategies
Other scenarios
0
1000
2000
3000
4000
5000
6000
7000
0 1 2 3 4 5 6
Month
Cu
mu
lati
ve n
um
ber
of
un
its
Section 1
Comparing strategies
comparing strategies
0
100
200
300
400
500
600
700
Chase Level Mixed
strategy
tota
l co
st
(th
ou
san
ds)
Hiring/firing costsInventory costs
Mathematical programming formulations
Cost definitions and input dataCH = Cost of hiring one workerCF = Cost of firing one workerCI = Cost of holding one unit of stockCR = Cost of producing one unit in regular timeCO = Cost of producing one unit in over timeCU = Idle cost per unit of productionCS = Cost to subcontract one unit of productionnt = Number of production days in period tK = Number of aggregate units produced by one worker in one
dayI0 = Initial inventoryW0= Initial workforceDt = Forecast of demand in period t
Problem variablesWt = Workforce level in period t
Pt = Number of units produced in period t
It = Inventory level at the end of period t
Ht = Number of workers hired in period t
Ft = Number of workers fired in period t
Ot = Overtime production in units
Ut = Worker idle time in units
St = Number of units subcontracted from outside
Constraints Conservation of workforce
Wt = Wt-1+ Ht – Ft, for all t=1,..,T Inventory balance
It = It-1+Pt+St-Dt, for all t=1,..,T Relationship between production and
workforcePt = K nt Wt + Ot - Ut, for all t=1,..,T
Constraints Initial levels for inventory and workforce
I0=I0, W0= W0
Ending levels for inventory and workforceIT=IT, WT= WT
Non-negativity constraintsWt, Pt, It, Ht, Ft, Ot, Ut, St >=0 for all t=1,..,T
Linear program
TtPWSUOIFH
TtDSPII
TtUOWKnP
TtFHWW
ScUcOcPcIcFcHc
tttttttt
ttttt
ttttt
tttt
tStUtOtRtItF
T
ttH
1 allfor 0,,,,,,,
1 allfor
1 allfor
1 allfor
Subject to
)( Minimize
1
1
1
Properties of the optimal solution
If cF>=0 and cH>=0, there could be either hiring or firing in one period (if at all), not both. i.e., HtFt=0, for all t=1,..,T
If cO>=0 and cI>=0, there could be either overtime or idle time in one period (if at all), not both. i.e.,OtUt=0, for all t=1,..,T
IF THESE CONSTRAINTS ARE INCLUDED INTO THE MODEL, IT BECOMES NON-LINEAR. FROM OPTIMALITY CONDITIONS, NOT NECESSARY TO INCLUDE INTO THE MODEL.EXTREME POINTS OF THE FEASIBLE REGION SATISFY THESE CONDITIONS
Properties of the optimal solution
Not necessarily integers- fractional numbers may not make sense for some variables, eg # of people hired/fired.
Requires an integer-linear programming model, imposing that variables are integers. Hovewer this is computationally difficult.
Solving as LP and then rounding may be possible. However, rounding must be performed carefully( may lead to infeasibilities).
Round to the next larger integer!
Extensions Adding buffers for uncertainty
It >= Bt, for all t=1,..,T
Bt= Buffer stock for period t defined in advance
Storage constraints for inventoryIt <= Vt, for all t=1,..,T
Vt=Storage capacity in period t
Extensions Limits on overtime
Ot<=Mt, for all t=1,..,T
Mt: maximum overtime in period t Capacity constraints on production
Pt<=Ct, for all t=1,..,T
Ct: capacity in period t
Allowing for backorders: Inventory level is the difference of +
and - inventory It=I+t-I-t , all non-negative for all periods
Holding cost: charged for + inventory Stockout cost: charged against –
inventory
Backorders and inventory In any period, there are either backorders or
positive inventory (if at all), but not both. i.e.,
TtII tt 1 allfor 0
Allowing for backorders
TtPWSUOIIFH
TtIII
TtDSPII
TtUOWKnP
TtFHWW
ScUcOcPcIcIcFcHc
ttttttttt
ttt
ttttt
ttttt
tttt
tStUtOtRtPtItF
T
ttH
1 allfor 0,,,,,,,,
1 allfor
1 allfor
1 allfor
1 allfor
Subject to
)( Minimize
1
1
1
Convex piecewise-linear costs
Linearization
TtPWSUOIFHH
TtDSPII
TtUOWKnP
TtHH
TtHHH
TtFHWW
ScUcOcPcIcFcHcHc
ttttttttt
ttttt
ttttt
t
ttt
tttt
tStUtOtRtItF
T
ttHtH
1 allfor 0,,,,,,,,
1 allfor
1 allfor
1 allfor *
1 allfor
1 allfor
Subject to
)( Minimize
21
1
1
21
1
12211
# of workers hired upto H*
Example-1 The Paris Paint Company is in the process of planning labor force
requirements and production levels for the next 4 quarters. The marketing department has provided production with the following forecasts of demand for Paris Paint over the next year.
Quarter Demand forecast in (1000s of gallons)
1 380
2 630
3 220
4 160
Assume that there are currently 280 employees with the company. Employees are hired for at least one full quarter. Hiring costs amount to $1,200 per employee and firing costs are $2,500 per employee. Inventory costs are $1 per gallon per quarter. It is estimated that one worker produced 1,000 gallons of paint each quarter. Assume that Paris currently has 80,000 gallons of paint in inventory and would like to end the year with an inventory of at least 20,000 gallons.Formulate the problem as an LP.
Linear Program-1
TtPWIFH
DDDD
WI
I
TtDPII
WP
TtFHWW
IFH
ttttt
tttt
tt
tttt
tt
T
tt
1 allfor 0,,,,
160000,220000,630000,380000
280,80000
20000
1 allfor
1000
1 allfor
Subject to
)25001200( Minimize
4321
00
4
1
1
1
Example 2 Sun Microsystems is the producer of computer workstations. For the year 2003,
they estimate their quarterly demand for high-end workstations to be the following:
Q1 5000
Q2 5000
Q3 9000
Q4 8000
Sun Microsystems focuses on innovation and design rather than manufacturing. Therefore, Sun developed strategic relationships with contract manufacturers Solectron and Celestica; Solectron promising 6000, Celestica promising 2000 units of maximum manufacturing capacity per quarter for Sun.Solectron is the preferred vendor for Sun and charges $1000 for each high end workstation that it manufactures. Sun may chose to use any amount of the capacity that Solectron promised without paying any penalties. Celestica, on the other hand, is the secondary vendor for Sun. It charges $1100 for each high-end workstation. Celestica requires that Sun declares the Q1 subcontracted amount in advance. Also, Sun can increase the subcontracted amount each quarter only if it pays $100 per unit of increase. In addition, Sun can never reduce the amount it subcontracted from Celestica from quarter to quarter.Inventory holding costs are $50 per quarter per unit.How much should Sun subcontract from Solectron and Celestica each quarter?
Linear Program-2
0,
1 allfor 0,,,,
8000,9000,5000,5000
0
1 allfor 2000
1 allfor 6000
1 allfor
1 allfor
1 allfor
Subject to
)5010011001000( Minimize
200
221
4321
0
2
1
1
21,22
21
221
1
SI
TtCSSSI
DDDD
I
TtS
TtS
TtDSII
TtCSS
TtSSS
ICSS
ttttt
t
t
tttt
ttt
ttt
ttt
T
tt
Example-3 The personal department of the A&M Corporation wants to
know how many workers will be needed each month for the next 4 month production period. The demands would be 1250, 1100, 950, 900 in months July, August, September, October.
The inventory on hand at the end of June was 500 units. The company wants to maintain a minimum inventory of 300 units each month. Each unit requires 5 employee hours. There are 20 working days each month, and each employee works an 8-hour day. The workforce at the end of June was 35 workers.
The workers can also work overtime, but overtime cannot exceed 30% of the regular time in each month. Overtime costs an additional $20 per unit produced.
Hiring costs $2500 per employee, firing costs $4000 per employee, payroll costs $3000 per employee per month, and inventory holding cost is $100.
Formulate the problem as an LP and solve
Linear Program-3
TtOPWIFH
DDDD
WI
TtI
TtDPII
TtWO
TtWP
TtFHWW
OIWFH
tttttt
t
tttt
tt
tt
tttt
tttt
T
tt
1 allfor 0,,,,,
900,950,1100,1250
35,500
1 allfor 300
1 allfor
1 allfor 6.9
1 allfor O 32
1 allfor
Subject to
)20100300040002500( Minimize
4321
00
1
t
1
1
Example 4- Component availability constraints Seagate is a manufacturer of hard drives for personal and
enterprise use. The enterprise division is trying to plan its production for the first six months in 2003. The forecast for these 6 months are given below
January 12000
February 14000
March 15000
April 16000
May 16500
June 16000
Seagate does not have any constraints on workforce or equipment for manufacturing hard drives. However, the enterprise hard drive requires two counts of a specific chip (application specific integrated circuit – ASIC) which is sourced from Texas Instruments. TI’s capacity is fixed for the first 4 months and it can allocate only a portion of its capacity to Seagate. This amounts to 26000 such chips in each of the first 4 months in 2003. Seagate may also choose to use subcontractors for manufacturing the hard drives which would cost an additional (on top of its own manufacturing) $50 per drive.If the inventory holding costs are $20 per unit per month for the hard drives and $5 per unit per month for the ASICs, find the optimal production plan for the hard drives and optimal purchase plan for the hard drives and ASICs.
Linear Program-4
61 allfor 0,,,,
16000,16500,1600015000,14000,12000
0
41 allfor 26000
61 allfor 2
61 allfor
Subject to
)50520( Minimize
554
321
00
1
1
6
1t
tSPPII
DDDDDD
II
tP
tPPII
tDSPII
SII
tc
th
tct
ht
ch
ct
ht
ct
ct
ct
tth
tht
ht
tc
th
t
Production Planning problems with concave costs
Why concave costs? Economies of scale Setup costs associated with
production, subcontracting, overtime, alternate resources
Cannot be modeled as linear programs
May be more difficult to solve
Modeling fixed (setup) costs Assume there is a fixed cost associated with
subcontracting. No idle time or overtime allowed.
largely sufficient
)1,0(
1 allfor 0,,,,,
1 allfor
1 allfor
1 allfor
Subject to
)( Minimize
1
1
1
M
y
TtPWSIFH
MyS
TtDSPII
TtWKnP
TtFHWW
yKScPcIcFcHc
t
tttttt
tt
ttttt
ttt
tttt
tStStRtItF
T
ttH
Example 5- Component availability constraints Seagate is a manufacturer of hard drives for personal and
enterprise use. The enterprise division is trying to plan its production for the first six months in 2003. The forecast for these 6 months are given below
January 12000
February 14000
March 15000
April 16000
May 16500
June 16000Seagate does not have any constraints on workforce or equipment for manufacturing hard drives. However, the enterprise hard drive requires two counts of a specific chip (application specific integrated circuit – ASIC) which is sourced from Texas Instruments. TI’s capacity is fixed for the first 4 months and it can allocate only a portion of its capacity to Seagate. This amounts to 26000 such chips in each of the first 4 months in 2003. Seagate may also choose to use subcontractors for manufacturing the hard drives which would cost an additional (on top of its own manufacturing) $10 per drive. In addition, there is a fixed cost of $20,000 working with a subcontractor in any month.If the inventory holding costs are $20 per unit per month for the hard drives and $5 per unit per month for the ASICs, find the optimal production plan for the hard drives and optimal purchase plan for the hard drives and ASICs.
Mixed Integer Program
61 allfor 0,,,,
16000,16500,1600015000,14000,12000
0
61 allfor )1,0(
61 allfor 89500
41 allfor 26000
61 allfor 2
61 allfor
Subject to
)2000010520( Minimize
554
321
00
1
1
6
1t
tSPPII
DDDDDD
II
ty
tyS
tP
tPPII
tDSPII
ySII
tc
th
tct
ht
ch
t
tt
ct
ht
ct
ct
ct
tth
tht
ht
ttc
th
t
From Aggregate Plan to Master Production Schedule
The result of the aggregate plan is the production quantities, inventory levels and required resources at the aggregate level in the mid term.
The firms are expected to act (acquire workforce & resources, contact suppliers) based on the aggregate plan.
Such actions will be input (constraints) to lower level (i.e. more detailed and shorter term) decisions in the company
Consistency between the aggregate plan and the master production schedule is desired, but not always possible
Initial master production schedule dictated by the demand plan (forecast) may not always be feasible.
Disaggregation into MPS X*=optimal # of aggregatee units to
be produced for a product group , found by LP solution
This aggregate qty should be disaggregated into individual products within this product group
Disaggregation into MPS Min Kj j/Yj , j=1.....J
s.t. Yj=X* j=1.....J
aj<=Yj<=bj Kj= setup for item j within the familyj = annual demand for item j
aj,bj: lower/upper bounds for item jKj j/Yj: average annual setupcost for item j
Inputs to Master Production Schedule
Forecast by end item (sometimes also by customer class or distribution channel) usually weekly sometimes monthly
For each item, manufacturing/distribution process flow
For each stage of manufacturing/distribution process flow
Consumption rate of critical components Consumption rate of critical resources
Availability for critical components and resources
Lead times (for time phasing) Prioritization and allocation schemes for constrained
situations
MPS
Links tactical and operational planning stages Engine that drives MRP, CRP, purchasing and shop
floor execution systems Anticipated build schedule for end-products or
major product options A statement of production, not demand Sales forecast is an input but MPS is not a forecast Always stated in terms of part numbers for which
BOM exists After constructing the MPS we make a rough-cut
capacity planning in terms of critical work centers
Creating MPS Decide on appropriate planning horizon Estimate requirements for each period in
terms of end-products in the planning horizon Account for any backorders Compare requirements against actual and
projected inventory balance Build the MPS Use rough-cut capacity planning to evaluate
feasibility If not, revise as appropriate
Output: a matrix in which rows showing individual products, columns showing time periods